5) A spherical storage tank has a radius of 3 feet, and half of it is underground: The equation of the curve in this drawing is x2+y²=9. The tank is full of a fluid having a weight density of w . How much Ib work is required to pump all the fluid up to the top of the tank? [The answer will be an expression containing w.]

Q: 3. Find the right triangle of largest area inscribed under the parabola y = 4 – r² with its base…

A: Given: y=4-x2 Vertex of the parabola will be at 0,4 and the points where parabola intersect the…

Q: Draw the isoclines with their direction markers and sketch several solution curves, including the…

A: Given  dydx=y→       1             dyy=dx          ln  y=x+C            y=C1  ex     →        2…

Q: The centroid of the half-ellipse-shaped-cross-sectional area shown is given by: 4 Ye 9 - 25m9-y² dy

A: Given half ellipse shaped cross section is:…

Q: 1. Find the equation of the tangent plane to the given surface at the point (0,-1,-2). * 2r2 „2

A: by using formula of plane

Q: The surface of a mountain is modeled by the equation h(x, y) = 5000 − 0.001x2 − 0.004y2. A mountain…

A: Hints:  The given equation is h(x,y)=5000-0.001x2-0.004y2 and the given point is (500,500,3750). To…

Q: If the length of the curve y = 3 x7 from the point (0, 0) to the point (xo, f(xo)) is L = 8 , then…

A: Length of curve

Q: 1 Find the exact length of the curve: r(t) = , 6.

A:

Q: write the equationns of the curves enclosing a reigon such that when it is rotated aboutthe x-axis,…

A:

Q: Suppose surface #1 is defined by o =45° , 0 <ps9, 0 < z< 6 and surface #2 is defined by 0 =56.3° The…

A: given:

Q: A] Find the length of the arc of the semi-cubical parabola y2 x (1,1) and (5,5V5). 8250 between the…

A: Solution:Given        y2=x3    between the points (1,1) and (5,55)To find the Arc…

Q: 1. Find an equation of the tangent plane to the given surface at the indicated point. 2² + y? – 3z =…

A:

Q: Find the exact length of the curve. y= In{1 – x²), o s 1. 0 sxs! 7. I Ju =

A: We will find the exact length of the curve as shown below

Q: Consider that the geometry of the arc for billboard for the pool is following the hyperbolic…

A:

Q: The area of a cylinder sector with radius r and central angle of radian measure 0 is A = r20, and…

A: If the area of the sector is 1 sq. unit, then A=112r2θ=1θ=2r2          ...(1)

Q: Now, we can say that the work done in stretching the spring 15 in. = X ft beyond its natural length…

A: Convert the length 9 inch to feet. Then use the Hooke’s Law to calculate the value for k.

Q: Find the length of the spiral r = eθ/√2 from θ = 0 to θ = π.

A: Given Differentiating r with respect to thita ,we get

Q: for the given audratic equation determine, i- which way the parabolla opens ii- the cordinate of…

A:

Q: Consider that the geometry of the arc for the billboard for the pool is following the hyperbolic…

A: As per bartleby guidelines, for more than one questions asked only one is to be answered,kindly…

Q: What is the shortest distance from the surface xy +12x +z^2 =144 to the origin?

A: To find the shortest distance from surface xy +12x +z2 =144  to origin.

Q: 1. Find the absolute minimum value of 0 < x < 4 f (x, y) = 4x² + 2xy – 2y? – 18x on the rectangular…

A:

Q: The area of a triangle is to be computed from the formula A = ½ab sinθ, where a and b are the…

A: Given that, the area of the triangle is determined by A=12absinθ where a, b are two sides and θ is…

Q: 26. A cross section of the parabolic mirror in a car headlight is modeled by y = 12x. Where should…

A: Given that a cross section of the parabolic mirror in a car headlight is modeled by y2=12x. Clearly,…

Q: Consider the hyperbolic PDE shown below = 4- at? For a value of At=0.1, what are the values of Ax…

A:

Q: To find the points on the cone z? = x² + y² that are closest to the point (5,2,0) , we need to solve…

A: Given  z2=x2+y2 let the P (5,2,0) and P (x,y,z) so, d = x-52+y-22+z-02 d=x-52+y-22+z2d=…

Q: Consider the following.x = et - 6y = e2t(a) Eliminate the parameter to find a Cartesian equation of…

A: From the equation for x, write t in terms of x.

Q: If the infinite curve y = e-2x, x 2 0, is rotated about the x-axis, find the area of the resulting…

A:

Q: 4. Find the equation of the tangent plane at the point (-2, 1, -3) to the ellipsoid 22 + y? + = 3 4

A:

Q: The equation of a curved road is given as y = 100+6x–0.015x² +0.0000111? Determine the places where…

A: The equation of the curved road given is: y=100+6x-0.015x2+0.00001111x3 Measure of the "Sharpness of…

Q: sphere will reach point B as it swings about 'O' (a) If OA is a rope vertical plane. Determine the…

A: Answer:

Q: n the given question as follow, find the work done in pumping gasoline that weighs 42 pounds per…

A: We need following equations to find the required work done :               W = ∫ab F dx ..... (1)…

Q: 2.lexPress the faiauing curve inthe form of Yara meter eQuation al Y=1-×モン-2 b.) y? zzg,x>4

A: Solution given below

Q: 1. Find the exact length of the curve. (a) y² = 4(x+ 4)°, 00 (b) y = In z, 1<I<2

A:

Q: stuck on the attached problem

A: Please refer to the white board. We have to find the volume shaded in the first diagram. This volume…

Q: 4. Consider the equation of the hyperbola given by x2 + 2xy – y² = 7. Find an equation of the…

A:

Q: drawn from the point A(2,10) on the parabola y = x^2 + bx + c Since the tangent is parallel to y =…

A:

Q: . (a) Show that as (x, y)→ (0, 0) along any straight line y = mx, or along any parabola y = kx², the…

A: Given : f(x) = x³y/(2x^6+y²)  To find : To prove that limit exist along parabola

Q: A circular cylinder of radius 1 and the parabolic cylinder x = 2y? – 1 2 =

A: In this question, we have to find the curve of intersection.

Q: (b) Consider the cycloid C parametrized by x = t – sin t and y = 1 – cos t, for all real t. Find…

A: Solution x= t-sint, y =1-cost C(0) = (0,0) C(pi) = (pi-sin pi, 1-cos pi) = (pi, 2) C(2pi) = (2pi-0,…

Q: To find the points on the cone z2 = x² + y? that are closest to the point (6, 3,0) , we need to…

A: Proceed as shown below

Q: 7.38. Calculate the second moment of area for a 2 × 6 piece of lumber about the x-x and y-y axes, as…

A:

Q: What is the shortest distance from the surface xy+3x+z^2=12 to the origin?

A: Let (x ,y, z) any point on surface. We have to minimize function f(x,y,z) = x2 + y2 + z2 under…

Q: A parabola has an equation of y²=2x, compute the radius of curvature at point (2, 2)

A:

Q: Draw the isoclines with their direction markers and sketch several solution curves, including the…

A: According to the given information, it is required to draw the isoclines with direction makers and…

Q: In a plane, l lm and t 1 m. l and t related? By appearance, how are

A: We know that in a plane if l⊥m and t⊥m then l will be parallel to t So, l and t are related as l∥t

Q: If x represents the average velocity of a plane in still air and y represents the average velocity…

A: If x represents the average velocity of a plane in still air and y represents the average velocity…

Q: A water tank has the shape obtained by revolving the curve y = x^4/3 around the y-axis. What is the…

A:

Q: Find the diffrential equation of for each of the curves determined by the condition that at each…

A: length of subtangent =ydydx

Q: 7.) Find an equation of the tangent plane to the surface, x y(2z-3), at the point P(4,4,2)

A:

Q: If the two spheres (x-4)^2+(y-1)^2+z^2=27 and (x-13)^2+(y-10)^2+(z+9)^2=a are externally tangent to…

A: Consider the given two spheres x-42+y-12+z2=27 and x-132+y-102+z+92=a are externally tangent to each…

Q: The figure shows a semicircle with radius 1, horizontal diameter PQ, and tangent lines at P and Q.…

A: As per the question we are given a circle of radius 1 and a horizontal line above the x-axis. Now we…

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.